Algebra is hard
For certain values of k and m, the system
3a + 2b = 2
6a + 2b = k + 3a + mb
has infinitely many solutions (a,b). What are k and m?
We can only have infinitely many solutions (a,b) when both of our equations are identical. We have:
3a + 2b = 2
6a + 2b = k + 3a + mb
If we subtract 3a from both sides of our lower equation we get
3a + 2b = k + mb
Now, the left-hand side of both equations is identical.
For the right-hand side, we can set:
k+mb=2
We can see that m must be 0, as if it is not we will have a variable with b, so our result will vary based on b, and not always be 2. This cannot be so we need m=0 as it will cancel out b. Then, we are left with k+0(b)=2 k=2.
Hence, we have that m=0 and k=2.
I hope this helped!
